# Torsion tensor

From Encyclopedia of Mathematics

A tensor of type that is skew-symmetric with respect to its indices, obtained by decomposing the torsion form of a connection in terms of a local cobasis on a manifold . In particular, in terms of a holonomic cobasis , , the components of the torsion tensor are expressed in terms of the Christoffel symbols (cf. Christoffel symbol) of the connection as follows:

#### Comments

In terms of covariant derivatives and vector fields , the torsion tensor can be described as follows:

#### References

[a1] | N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) |

[a2] | D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) |

[a3] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |

**How to Cite This Entry:**

Torsion tensor. M.I. Voitsekhovskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Torsion_tensor&oldid=14358

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098